|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.20.10.0001.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Hypergeometric1F1[a, b, z] ==
1 + (a (z/b))/(1 + -(((1 + a) z)/(2 (1 + b)))/(1 + ((1 + a) z)/(2 (1 + b)) -
((2 + a) z)/(3 (2 + b))/(1 + ((2 + a) z)/(3 (2 + b)) -
((3 + a) z)/(4 (3 + b))/(1 + ((3 + a) z)/(4 (3 + b)) -
((4 + a) z)/(5 (4 + b))/(1 + ((4 + a) z)/(5 (4 + b)) +
\[Ellipsis])))))
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric1F1", "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["a", " ", RowBox[List["z", "/", "b"]]]], ")"]], "/", RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]]]]], "/", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]]], "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", "z"]], RowBox[List["3", " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", "z"]], RowBox[List["3", " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]], "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a"]], ")"]], " ", "z"]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["3", "+", "b"]], ")"]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a"]], ")"]], " ", "z"]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["3", "+", "b"]], ")"]]]]], "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "+", "a"]], ")"]], " ", "z"]], RowBox[List["5", " ", RowBox[List["(", RowBox[List["4", "+", "b"]], ")"]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "+", "a"]], ")"]], " ", "z"]], RowBox[List["5", " ", RowBox[List["(", RowBox[List["4", "+", "b"]], ")"]]]]], "+", "\[Ellipsis]"]]]]]]]]]]], ")"]]]]]], ")"]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ; </mo> <mi> b </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> z </mi> <mo> / </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> … </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mstyle> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric1F1 </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <ci> a </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric1F1", "[", RowBox[List["a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["1", "+", FractionBox[RowBox[List["a", " ", "z"]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["3", " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", "z"]], RowBox[List["3", " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List["3", "+", "b"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a"]], ")"]], " ", "z"]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["3", "+", "b"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "+", "a"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["5", " ", RowBox[List["(", RowBox[List["4", "+", "b"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "+", "a"]], ")"]], " ", "z"]], RowBox[List["5", " ", RowBox[List["(", RowBox[List["4", "+", "b"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HypergeometricPFQ[{},{},z] | HypergeometricPFQ[{},{b},z] | HypergeometricPFQ[{a},{},z] | HypergeometricPFQ[{a1},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1},z] | HypergeometricPFQ[{a1,a2},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
|
|
|